Extended differential symbol and the Kato homology groups

TL;DR

This paper explores the relationship between higher Chow groups and Kato homology groups for varieties over various fields, extending previous work on differential symbols and reciprocity sheaves.

Contribution

It introduces an analogue of the differential symbol map in the context of Kato homology and investigates its structure over different types of arithmetic fields.

Findings

Higher Chow group corresponds to 0-th Kato homology group for certain varieties.

Structural insights into 0-th Kato homology over finite, local, and global fields.

Results expressed in terms of reciprocity sheaves.

Abstract

Building on our previous work, we investigate an analogue of the differential symbol map used in the Bloch-Gabber-Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th Kato homology group. Inspired by Akhtar's theorem on higher Chow groups, we investigate the structure of the 0-th Kato homology group for varieties over arithmetic fields, including finite fields, local fields, and global fields of positive characteristic. We also express our results in terms of reciprocity sheaves.